Global asymptotic stability for Gurtin-MacCamy's population dynamics model

TL;DR

This paper establishes the global asymptotic stability of an age-structured population model with nonlinear birth function, using Lyapunov functionals, and discusses the potential for Hopf bifurcations near the stability region.

Contribution

It provides a rigorous proof of global stability for Gurtin-MacCamy's model with a Ricker-type birth function, including conditions for bifurcation analysis.

Findings

Proved global asymptotic stability of the positive equilibrium.

Established existence of global attractors for the model.

Identified conditions under which Hopf bifurcation may occur.

Abstract

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker's type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semi-flow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.